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~2[Ll = a (o<p 2 in .NET
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4m 2 (835) s=t=u= This is clearly in agreement with the lowestorder value r(4)[O] =  A. Analogous normalization conditions can and must be introduced in any renormalizable theory. Of course, the foregoing is by no means a proof that divergences may be disposed of by counterterms. In particular, we have not completely proved that w(r(2)) = 2 implies that has the behavior (831). Indeed, we are not really in the conditions of the corollary (829), since all subdiagrams are not necessarily superficially convergent, but may have been renormalized by lowerorder counterterms. That our reasoning is nevertheless correct will actually follow a posteriori, when we will have shown that this procedure does lead to a finite renormalized theory. Here we only want to recall the logic of the method and stress the necessity of normalization conditions. At this point, it may be worth recalling that the difference between renormalizable and nonrenormalizable theories lies in the number of these conditions. While in the former a finite number of conditions suffice to define the theory in terms of a finite number of renormalized parameters, the renormalization of the latter requires an infinite set of such conditions; ultimately, a nonrenormalizable theory will depend on an infinite number of parameters. There is a great amount of arbitrariness in the choice of the normalization conditions. The only proviso is that they must be satisfied to lowest order, so as to fix unambiguously the subtractions to higher orders. We shall reexamine this point in Sec. 825 below. Owing to this arbitrariness, it may be more convenient to use a less physical, intermediate renormalization. When all the fields have a nonvanishing mass, it is safe to choose normalization conditions at the origin in momentum space. In the above example of the <p4 theory, the conditions will read r;;i
r (2)(P2) Ip2=O = R
r}i)(o, 0, 0, 0) = m 2
(836) The quantity m as defined by (836) is no longer the physical mass, even though it is related to it. When dealing with particles of nonzero spin, the tensor structure of Green's functions must be taken into account. It may happen that only the form factors of some tensors are divergent (for instance, in the study of the vertex function 388 QUANTUM FIELD THEORY
in the last chapter, F2 was found finite and Fl divergent). Only the latter requires subtractions and normalization conditions. A consequence of the previous recursive construction of the counter terms concerns their structure. In a renormalizable theory, the counterterms satisfy the criterion of renormalizability. This is what we found in spinor electrodynamics to the oneloop approximation, where the only counterterms had the form Similarly, in the <p4 theory, the counterterms are: (0<p)2:, : <p2:, : <p4: of degree less or equal to four. In scalar electrodynamics, the situation is slightly different, since starting from the monomials generated by minimal coupling we find counterterms of the same structure, plus a new term of the type : (<p t <p :. Therefore, we end up with a mixture of electrodynamics and <p4 selfcoupling, but the theory remains renormalizable. This also means that scalar electrodynamics depends on a supplementary unexpected parameter, namely, the value of the fourpoint function at a given point. More generally, if a proper diagram G of a renormalizable theory is superficially divergent, 0::; w(G) ::; 4  iEF  EE  () [we use Eq. (818), where Wv ::; 4J, the corresponding counterterm has EF fermion fields, EE boson fields, and a number () of derivatives. Therefore, its dimension, in the sense of Eq. (817) is (837) The counterterm thus generated has a dimension less or equal to four; hence it is also renormalizable. Whenever the counterterms have the same structure as monomials of the initial lagrangian, they may be considered as redefining the parameters of the theory. The quantities appearing in the lagrangian, resulting from the addition of counterterms to the initial monomials, will be referred to as bare parameters. The bare parameters are determined order by order in perturbation theory as functions of the renormalized quantities, so that the renormalization conditions are satisfied. We discussed this construction in the previous chapter for electrodynamics (see also Sec. 84 below). In the case of the <p4 theory, we write the lagrangian plus its counterterms as + ~2' =

